A new young type inequality involving Heinz mean

In this paper, we gave a new Young type inequality and the relevant Heinz mean inequality. Furthermore, we also improved some inequalities with Kantorovich constant or Specht?s ratio. Meanwhile, on the base of our scalars results, we obtain some new corresponding operator inequalities and matrix versions including Hilbert-Schmidt norm, unitarily invariant norm and related trace versions, which can be regarded as the application of our scalar results.

Refinements of the heinz inequalities for operators and matrices

Suppose that A, B ∈ 𝔹(𝓗) are positive invertible operators. In this paper, we show that $$\begin{array}{} \displaystyle A \# B \leq \frac{1}{1-2\mu}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}\\ \displaystyle\qquad~\,\leq\frac{1}{2}\bigg[ A \# B +H_\mu (A,B)\bigg]\\ \displaystyle\qquad~\,\leq\frac{1}{2}\bigg[ \frac{1}{1-2\mu}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}+H_\mu (A,B)\bigg]\\ \displaystyle\qquad~\,\leq \dots \leq \frac{1}{2^n}A \# B + \frac{2^n-1}{2^n}H_\mu (A,B)\\ \displaystyle\qquad~\,\leq \frac{1}{2^n(1-2\mu)}A^\frac{1}{2}F_\mu(A^\frac{-1}{2}BA^\frac{-1}{2})A^\frac{1}{2}+\frac{2^n-1}{2^n}H_\mu (A,B)\\ \displaystyle\qquad~\,\leq \frac{1}{2^{n+1}} A \# B +\frac{2^{n+1}-1}{2^{n+1}}H_\mu (A,B)\\ \displaystyle\qquad~\,\leq \dots \leq H_\mu (A,B) \end{array}$$ for each $\begin{array}{} \displaystyle \mu \in [0,1]\smallsetminus\{\frac{1}{2}\}, \end{array}$ where Hμ (A, B) and A#B are the Heinz mean and the geometric mean for operators A, B, respectively, and $\begin{array}{} \displaystyle F_{\mu}\in C({\rm sp}(A^\frac{-1}{2}BA^\frac{-1}{2})) \end{array}$ is a certain parameterized class of functions. As an application, we present several inequalities for unitarily invariant norms.

Some improvements of the young mean inequality and its reverse

The main objective of the present paper, is to obtain some new versions of Young-type inequalities with respect to two weighted arithmetic and geometric means and their reverses, using two inequalities $$\begin{array}{} \displaystyle K\Big(\frac{b}{a},2\Big)^r\leq\frac{a\nabla_{\nu}b}{a\sharp_{\nu}b}\leq K\Big(\frac{b}{a},2\Big)^R, \end{array}$$ where r = min{ν, 1 – ν}, R = max{ν,1 – ν} and K(t,2) = $\begin{array}{} \displaystyle \frac{(t+1)^2}{4t} \end{array}$ is the Kantorovich constant, and $$\begin{array}{} \displaystyle e(h^{-1},\nu)\leq\frac{a\nabla_{\nu}b}{a\sharp_{\nu}b}\leq e(h,\nu), \end{array}$$ where h = max $\begin{array}{} \displaystyle \{\frac{a}{b},\frac{b}{a}\} \end{array}$ and e(t,ν) = exp (4ν(1 – ν)(K(t,2)–1) $\begin{array}{} \displaystyle (1-\frac{1}{2t})\big). \end{array}$ Also some operator versions of these inequalities and some inequalities related to Heinz mean are proved.